Final answer:
To find the equation of the tangent line at the point (1,3) for the function f(x), compute the derivative to get the slope, 5, at that point. Then use the point-slope form, yielding the final equation y = 5x - 2.
Step-by-step explanation:
To find an equation of the tangent line to the graph of the function f(x) = x² + 3x - 1 at the point (1,3), we first need to determine the slope of the function at that point. We do this by finding the derivative of the function, f'(x) = 2x + 3, and evaluating it at x=1. The slope of the tangent line is f'(1) = 2(1) + 3 = 5.
The equation of a straight line is typically written as y = mx + b, where m is the slope and b is the y-intercept. Since we have the slope and a point on the line, we use the point-slope form to get the equation of the tangent line: y - y₁ = m(x - x₁). Plugging in our point (1,3) and the slope 5, we get y - 3 = 5(x - 1). Simplifying, we get the equation of the tangent line to be y = 5x - 2.